A detailed study of all the lattice equations derived
from the rhombic \(H^{4}\) 
family, including the construction of their three-leg forms,
Lax pairs, Bäcklund transformations and infinite hierarchies
of generalized symmetries,
has been presented in [1]}.
There are plenty of results about
the \(Q\)  and the rhombic \(H^{4}\)  equations.
On the contrary,
besides the CAC property
little is known about the integrability features
of the trapezoidal \(H^{4}\)  equations
and of the \(H^{6}\)  equations.
These equations where thoroughly studied
in a series of papers
[2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}
with some unexpected results.
In [2]} the algebraic entropy
[10]}, [11]}, [12]}, [13]}
of the trapezoidal \(H^{4}\)  and the \(H^{6}\)  equations was computed.
The result of this computation showed that the rate of growth
of all the trapezoidal \(H^{4}\) 
and of all \(H^{6}\)  equations is linear.
According to the algebraic entropy conjecture [14]}, [11]}
this fact implies the linearizability.
In [8]}, following the suggestions obtained in [3]}, [4]},
it was showed that the trapezoidal \(H^{4}\)  equations and all the \(H^{6}\)  equations are
Darboux integrable [19]}.
Finally in [7]} it was shown,
applying a modification of the procedure presented
in [21]},
that Darboux integrability provides the general
solutions of these equations.
Moreover in [4]} it was showed that
the three quad-equations found in [23]}
were Darboux integrable.
